Question 295.
Imagine a flat earth extending to infinity in all directions. How thick would the earth need to be to have the same gravity at the surface as we have now.
Assume a density of 5,515 kg/m³, g = 9.81 m/s².
Answer 295.
This question was answered by me using calculus and also by Rohan using Gauss's Law. Rohan's solution was selected as the best. Given here is the solution of this interesting question in the words of the asker, Frst Grade Rocks! Ω.
Both, Rohan and Frst Grade Rocks! Ω are my valuable contacts in Yahoo Answers! who keep asking such challenging questions that I am rarely able to answer.
Fred, a valuable contributor in YA! and in my contact list, made the following interesting observation on the value of g for the infinite flat surface of the earth.
"Note that for such an infinite, flat Earth, the acceleration of gravity would
= g throughout all space (outside Earth!), not diminishing at all with distance from either surface.
This is mathematically identical to that of finding the electric field of an infinite sheet of constant charge density."
Solution:
As per Gauss's Theorem, g ∝ M/SA,
where M = Mass and SA = Surface Area
For the earth,
M = Mass of Earth = 4/3 π ρ r³ and SA = Surface Area = 4 π r²
g.earth ∝ 1/3 ρ r ... (1)
For a flat infinite plane,
M = ρ d A, where d = depth of plane
and A = arbitrary cross sectional area.
SA = 2 * A
The factor of "2" is because you have a flux coming out
from both the sides of the plane, the bottom and the top
g.plane ∝ ρdA / (2A)
=> g.plane ∝ 1/2 ρ d ... (2)
Set g.earth = g.plane
=> 1/3 ρ r = 1/2 ρ d
=> d = 2/3 r
Taking radius of the earth = 6 378.1 km
d = 4252 km.
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